Found problems: 85335
2004 Croatia Team Selection Test, 3
A line intersects a semicircle with diameter $AB$ and center $O$ at $C$ and $D$, and the line $AB$ at $M$, where $MB < MA$ and $MD < MC.$ If the circumcircles of the triangles $AOC$ and $DOB$ meet again at $K,$ prove that $\angle MKO$ is right.
2013 National Olympiad First Round, 7
What is the sum of real roots of the equation $x^4-8x^3+13x^2 -24x + 9 = 0$?
$
\textbf{(A)}\ 8
\qquad\textbf{(B)}\ 7
\qquad\textbf{(C)}\ 6
\qquad\textbf{(D)}\ 5
\qquad\textbf{(E)}\ 4
$
1994 IMO Shortlist, 2
Let $ m$ and $ n$ be two positive integers. Let $ a_1$, $ a_2$, $ \ldots$, $ a_m$ be $ m$ different numbers from the set $ \{1, 2,\ldots, n\}$ such that for any two indices $ i$ and $ j$ with $ 1\leq i \leq j \leq m$ and $ a_i \plus{} a_j \leq n$, there exists an index $ k$ such that $ a_i \plus{} a_j \equal{} a_k$. Show that
\[ \frac {a_1 \plus{} a_2 \plus{} ... \plus{} a_m}{m} \geq \frac {n \plus{} 1}{2}.
\]
2018 Balkan MO Shortlist, C3
An open necklace can contain rubies, emeralds, and sapphires. At every step we can perform any of the following operations:
[list=1]
[*]We can replace two consecutive rubies with an emerald and a sapphire, where the emerald is on the left of the sapphire.[/*]
[*]We can replace three consecutive emeralds with a sapphire and a ruby, where the sapphire is on the left of the ruby. [/*]
[*]If we find two consecutive sapphires then we can remove them.[/*]
[*]If we find consecutively and in this order a ruby, an emerald, and a sapphire, then we can remove them.[/*]
[/list]
Furthermore we can also reverse all of the above operations. For example by reversing 3. we can put two consecutive sapphires on any position we wish.
Initially the necklace has one sapphire (and no other precious stones). Decide, with proof, whether there is a finite sequence of steps such that at the end of this sequence the necklace contains one emerald (and no other precious stones).
[i]Remark:[/i] A necklace is open if its precious stones are on a line from left to right. We are not allowed to move a precious stone from the rightmost position to the leftmost as we would be able to do if the necklace was closed.
[i]Proposed by Demetres Christofides, Cyprus[/i]
2016 Flanders Math Olympiad, 1
In the quadrilateral $ABCD$ is $AD \parallel BC$ and the angles $\angle A$ and $\angle D$ are acute. The diagonals intersect in $P$. The circumscribed circles of $\vartriangle ABP$ and $\vartriangle CDP$ intersect the line $AD$ again at $S$ and $T$ respectively. Call $M$ the midpoint of $[ST]$. Prove that $\vartriangle BCM$ is isosceles.
[img]https://1.bp.blogspot.com/-C5MqC0RTqwY/Xy1fAavi_aI/AAAAAAAAMSM/2MXMlwb13McCYTrOHm1ZzWc0nkaR1J6zQCLcBGAsYHQ/s0/flanders%2B2016%2Bp1.png[/img]
2003 India National Olympiad, 5
Let a, b, c be the sidelengths and S the area of a triangle ABC. Denote $x=a+\frac{b}{2}$, $y=b+\frac{c}{2}$ and $z=c+\frac{a}{2}$. Prove that there exists a triangle with sidelengths x, y, z, and the area of this triangle is $\geq\frac94 S$.
2017 Romania EGMO TST, P3
Determine all functions $f:\mathbb R\to\mathbb R$ such that \[f(xy-1)+f(x)f(y)=2xy-1,\]for any real numbers $x{}$ and $y{}.$
1996 Iran MO (3rd Round), 1
Let $a,b,c,d$ be positive real numbers. Prove that
\[\frac{a}{b+2c+3d}+\frac{b}{c+2d+3a}+\frac{c}{d+2a+3b}+\frac{d}{a+2b+3c} \geq \frac{2}{3}.\]
2005 Federal Competition For Advanced Students, Part 1, 3
For 3 real numbers $a,b,c$ let $s_n=a^{n}+b^{n}+c^{n}$.
It is known that $s_1=2$, $s_2=6$ and $s_3=14$.
Prove that for all natural numbers $n>1$, we have $|s^2_n-s_{n-1}s_{n+1}|=8$
2001 India IMO Training Camp, 1
Complex numbers $\alpha$ , $\beta$ , $\gamma$ have the property that $\alpha^k +\beta^k +\gamma^k$ is an integer for every natural number $k$. Prove that the polynomial \[(x-\alpha)(x-\beta )(x-\gamma )\] has integer coefficients.
2014 Thailand TSTST, 1
Let $x, y, z$ be positive real numbers. Prove that $$4(x^2+y^2+z^2)\geq3(xy+yz+zx).$$
2012 International Zhautykov Olympiad, 2
A set of (unit) squares of a $n\times n$ table is called [i]convenient[/i] if each row and each column of the table contains at least two squares belonging to the set. For each $n\geq 5$ determine the maximum $m$ for which there exists a [i]convenient [/i] set made of $m$ squares, which becomes in[i]convenient [/i] when any of its squares is removed.
2009 Oral Moscow Geometry Olympiad, 2
A square and a rectangle of the same perimeter have a common corner. Prove that the intersection point of the diagonals of the rectangle lies on the diagonal of the square.
(Yu. Blinkov)
2008 Harvard-MIT Mathematics Tournament, 6
Determine all real numbers $ a$ such that the inequality $ |x^2 \plus{} 2ax \plus{} 3a|\le2$ has exactly one solution in $ x$.
2024 Euler Olympiad, Round 1, 10
Find all $x$ that satisfy the following equation: \[ \sqrt {1 + \frac {20}x } = \sqrt {1 + 24x} + 2 \]
[i]Proposed by Andria Gvaramia, Georgia [/i]
2017 Taiwan TST Round 2, 2
Let $ABC$ be a triangle such that $BC>AB$, $L$ be the internal angle bisector of $\angle ABC$. Let $P,Q$ be the feet from $A,C$ to $L$, respectively. Suppose $M,N$ are the midpoints of $\overline{AC}$ and $\overline{BC}$, respectively. Let $O$ be the circumcenter of triangle $PQM$, and the circumcircle intersects $AC$ at point $H$. Prove that $O,M,N,H$ are concyclic.
2015 Romania National Olympiad, 2
Show that the set of all elements minus $ 0 $ of a finite division ring that has at least $ 4 $ elements can be partitioned into two nonempty sets $ A,B $ having the property that
$$ \sum_{x\in A} x=\prod_{y\in B} y. $$
2010 Contests, 3
In an acute-angled triangle $ABC$, $CF$ is an altitude, with $F$ on $AB$, and $BM$ is a median, with $M$ on $CA$. Given that $BM=CF$ and $\angle MBC=\angle FCA$, prove that triangle $ABC$ is equilateral.
1989 Bundeswettbewerb Mathematik, 2
A trapezoid has area $2\, m^2$ and the sum of its diagonals is $4\,m$. Determine the height of this trapezoid.
2013 India PRMO, 13
To each element of the set $S = \{1,2,... ,1000\}$ a colour is assigned. Suppose that for any two elements $a, b$ of $S$, if $15$ divides $a + b$ then they are both assigned the same colour. What is the maximum possible number of distinct colours used?
2017 Taiwan TST Round 1, 2
Given $a,b,c,d>0$, prove that:
\[\sum_{cyc}\frac{c}{a+2b}+\sum_{cyc}\frac{a+2b}{c}\geq 8(\frac{(a+b+c+d)^2}{ab+ac+ad+bc+bd+cd}-1),\]
where $\sum_{cyc}f(a,b,c,d)=f(a,b,c,d)+f(d,a,b,c)+f(c,d,a,b)+f(b,c,d,a)$.
MBMT Team Rounds, 2015 F7 E4
Compute $\frac{x^2 + 8x + 7}{x^2 + 9x + 14}$, if $x = 2015$.
2011 IMO Shortlist, 3
Let $n \geq 1$ be an odd integer. Determine all functions $f$ from the set of integers to itself, such that for all integers $x$ and $y$ the difference $f(x)-f(y)$ divides $x^n-y^n.$
[i]Proposed by Mihai Baluna, Romania[/i]
2002 AMC 10, 7
If an arc of $ 45^\circ$ on circle $ A$ has the same length as an arc of $ 30^\circ$ on circle $ B$, then the ratio of the area of circle $ A$ to the area of circle $ B$ is
$ \textbf{(A)}\ \frac {4}{9} \qquad \textbf{(B)}\ \frac {2}{3} \qquad \textbf{(C)}\ \frac {5}{6} \qquad \textbf{(D)}\ \frac {3}{2} \qquad \textbf{(E)}\ \frac {9}{4}$
2002 Romania Team Selection Test, 3
Let $M$ and $N$ be the midpoints of the respective sides $AB$ and $AC$ of an acute-angled triangle $ABC$. Let $P$ be the foot of the perpendicular from $N$ onto $BC$ and let $A_1$ be the midpoint of $MP$. Points $B_1$ and $C_1$ are obtained similarly. If $AA_1$, $BB_1$ and $CC_1$ are concurrent, show that the triangle $ABC$ is isosceles.
[i]Mircea Becheanu[/i]